what's the name of the constant that emerges when dividing two sides of a triangle, which is equal for all similar triangles I know this is a simple question but I cant find the name for this.
(For a figure go to the bottom of the question)
I know that for similar triangles $\Delta ABC$ $\Delta A'B'C'$ I can divide any two similar sides and get the scale factor $K$
$\frac{AB}{A'B'}=\frac{BC}{B'C'}=\frac{AC}{A'C'}=K$
The scale, of course, changes for different similar triangles. However, if I where to do some algebra.
$\frac{AB}{A'B'}=\frac{AC}{A'C'}$
$AB=\frac{AC}{A'C'}A'B'$
$AB=\frac{A'B'}{A'C'}AC$
$\frac{AB}{AC}=\frac{A'B'}{A'C'}=m$
I get to some constant $m$ that does not change between any similar triangles.
I also noticed that has some kind of relationship with the law of sines.
I want to know:
1)what is the name of this constant or how can I search for it (could not find it).
2)If there is an analogus euclid element's proposition about it.
Here is an example on geogebra. If you drag the "drag me" point you are generating a diferent triangle $A'B'C'$ similar to $ABC$ and you can see how $K$ changes but $m$ stays the same for both triangles independently of their difference in size
https://www.geogebra.org/classic/kbwzynd6
 A: The constant $K$ is often called "similarity ratio" or "scale factor". The constant $m$ does not have a definite name (there are various constants of this type). As regards Euclid's propositions, the most important in this context are those from Book VI, from VI-4 to VI-7.
A: It doesn't have a name, unless the triangle is right-angled.
You probably noticed that there are six such constants: your $m = \frac{AB}{AC} =  \frac{A'B'}{A'C'}$, its brother $n = \frac{BC}{BA} =  \frac{B'C'}{B'A'}$ and its sister $p = \frac{CA}{CB} = \frac{C'A'}{C'B'}$ together with their inverses.
All six are the ratio between two sides meeting in a point and you don't have any more information than that to tell them apart. If I give you a random triangle (but do not tell you the name of the vertices) you have no way to say which one is $m$ which one is $m^{-1}$, which one is $n$ which one is $n^{-1}$, which one is $p$ and which one is $p^{-1}$.
That is not a very fortunate situation. The reason that these constants don't have a name is that if they had and I were to use that name you still wouldn't know what constant I was talking about.
